On the plains north and east of Mosul, far from the battle in the city centre, a new frontline is taking shape. Mounds of earth have been heaped above a trench gouged out of the ground along about 650 miles (1,050km) of northern Iraq, which before the war with Islamic State was in Arab hands.

The berm runs from Sinjar, in the north-west, to Khanaqin, near the Iranian border, following the line of Kurdish military control. Woven into it are peshmerga positions, and on top flies the Kurdish flag, a clear statement of the Kurds’ hope that their role in fighting the war has already secured them a bigger slice of Iraq.

As Iraqi forces have pushed further into central Mosul over the past week, ousting Isis from the university and reaching the Tigris river that divides the city, the Kurds have been putting the finishing touches on what officials in Erbil, the capital of Iraqi Kurdistan, call a military line that commandeers more land than they have ever had in the modern Iraqi state.

The trench and berm, the Kurds say, is a recognition of their role in securing the city’s eastern and northern outskirts in the first week of fighting, which started on 17 October last year and is now into its fourth month. Regional officials expect the battle for Mosul to continue for at least three more months, possibly into the summer, despite the renewed momentum of the Iraqi army. Across the river is the Grand Mosque where the Isis leader, Abu Bakr al-Baghdadi, proclaimed himself caliph of an Islamic state in July 2014.

Out of the fighting, the Kurds have turned their attention to advancing political goals. Consolidating military gains made by the peshmerga in and around the Christian towns of Bartella and Bashiqa have been central to the plans of senior leaders.

Fighting for every yard: on the ground with Kurdish troops pushing into Mosul

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After losing the Nineveh plains and almost losing Erbil as Isis rampaged towards them in August 2014, the Kurds have increased the land mass under their control by up to 40%. In the first week of fighting for Mosul, another 193 sq miles (500 sq km) was added. “We’re not moving from the frontlines,” said one Kurdish official. “Especially the hilltops such as Sinjar.”

Focus among the participants in the fight to recapture Mosul is starting to shift to what follows Isis’s seemingly inevitable defeat. Iraq’s weakened central government is hoping to restore its authority in Mosul, with the success of its military seen by many in government as a nation-building measure that could restore trust between the minority Sunnis, who are dominant in Mosul, and Iraq’s Shia Muslim majority, which fills the ranks of the military.

Ceding ground, or more authority, to the Kurds of the largely autonomous north has been strongly resisted by Baghdad, which has played a diminishing role in Kurdish affairs over the past decade. However, senior Kurdish officials say the end of the war should mark a time of reckoning.

The president of the Kurdish north, Masoud Barzani, last year failed to deliver a referendum he had pledged to hold in November, which he said would further move the area away from central government control. Facing domestic political paralysis and an economy almost solely dependent on oil, the sale of which Baghdad insists must be coordinated centrally, Barzani has hung much on the fate of the Isis war.

He and other senior officials are hoping that the shared burden of the status quo, along with the newly carved line in the dirt will give the Kurds leverage.

“A lot of Iraqi leaders understand deep down that it is gone, that it’s a lost cause,” said the chancellor of the Kurdish region security council, Masrour Barzani, of the concept of a unified Iraq. “The essence of this relationship should be one between Kurdistan and Iraqi Arabs, not a nationalistic approach, but a territorial relationship. We cannot live under the same formula anymore. We need to work out how we can be good neighbours.

“The line where we are right now is a military line, not a political line. This is the minimum outreach of Kurdistan. We are not going to compromise on anything we did prior to 17 October. Anything beyond that is subject to agreements and the will of the people in those areas. The trenches are not politically binding, but that does not mean we don’t have a say in what happens beyond those areas.”

A frontline outpost in Sinjar in October. Photograph: Alessandro Rota for the Guardian

The Kurds are hoping that in the postwar shakeout some villages beyond the new, nominal border may choose them over Baghdad, further increasing their hold in the Nineveh plains. They have also insisted that Sinjar, which was reconquered in a peshmerga-led offensive 15 months ago more than a year after they had surrendered it, will not be returned to Baghdad.

About 12 miles from the foothills of Mount Sinjar, which towers above the Yazidi town, Isis remains bunkered down in the towns of Ba’aj, Billij and Tel Afar. Not far away, Shia militias, a powerful component of the fight against the terror group, are preparing for the Iraqi army assault on Mosul west of the Tigris. To their west is the remaining heartland of Isis, which spills towards the Syrian border and on towards Raqqa, one of the group’s two remaining main centres of gravity.

“There is a lot in this fight for everyone,” said a senior Iraqi minister. “The Kurds are getting ahead of themselves as they often do. They always miscalculate. The spoils of war will be divided on many levels when the terrorists are defeated. Everyone will want their share.”